Time-Dependent Determinantal Point Processes

该建议侧重于研究随机矩阵型的时变离散和连续概率模型。更准确地说，我们使用依赖于时间的行列式点过程。根据模型的三个来源：相互作用粒子系统(完全不对称排斥过程、多核生长过程等)、大群表示理论(无限对称群、无限维酉群等)和Schur过程，它们可以被视为适用于各种计数组合学问题的非常一般的代数形式。尽管起源不同，但这些主题密切相关，并且它们方便地提供了关于一类含时行列点过程的三种不同的观点。这样的模型自然出现在数学和物理中，直到最近还没有工具来计算它们精细的大时间渐近性。几年前，统计物理、组合学、分析和表示理论的方法组合取得了惊人的成功：对于少数几个模型，我们不仅能够严格控制波动的大小，而且我们还找到了发挥著名钟形曲线作用的显式分布。我们计划将这一成功转化为一种方法，为更广泛的增长模型提供结果，同时扩大该学科与数学和数学物理其他领域的联系。

The proposal focuses on the study of time-dependent discrete and continuous probabilistic models of random matrix type. More exactly, we work with time-dependent determinantal point processes. We use three different approaches corrersponding to three sources of our models: Interacting Particle Systems (the totally asymmetric exclusion process, the polynuclear growth process, etc.), Representation Theory of Big Groups (the infinite symmetric group, the infinite-dimensional unitary group and the like), and Schur processes, which may be viewed as a very general algebraic formalism suitable for various problems of Enumerative Combinatorics.Despite different origins these themes are closely interrelated, and they conveniently provide three different points of view on the class of time-dependent determinantal point processes.The proposal focuses on the study of large time fluctuations of various models of random growth in one space dimension. Such models naturally appear in mathematics and physics, and until recently no tools were available for computing their fine large time asymptotics.A few years ago a combination of methods from statistical physics, combinatorics, analysis, and representation theory lead to a stunning success: For a handful of models not only we were able to rigorously control the size of the fluctuations but we also found explicit distributions that play the role of the famous bell-shaped curve. We plan to turn this success into a method that delivers results for a much wider class of growth models while broadening the connection of the subject to other domains of mathematics and mathematical physics.